PUNJAB PUBLIC SERVICE COMMISSION WRITTEN TEST FOR THE POST OF LECTURER IN MATHEMATICS 2015

Q.01 \fn_cm \int_{-4}^{0}\frac{t}{\sqrt{16-t^2}}dt=__________________

A.   0

B.   Divergent

C.   -4

D.   4

Check Answer
C
View Explanation
  \fn_cm \int_{-4}^{0}\frac{t}{\sqrt{16-t^2}}dt=-\frac{1}{2}\int_{-4}^{0}\frac{-2t}{\sqrt{16-t^2}}dt=-\sqrt{16-t^2}|_{-4}^{0}=-4 

Q.02  The period of function \fn_cm Acos\omega t+Bsin\omega t is

A.  \fn_cm \frac{\omega}{2\pi}

B.  \fn_cm 2\pi \omega

C.  \fn_cm \frac{2\omega}{\pi}

D.  \fn_cm \frac{2\pi}{\omega}

Check Answer
D
View Explanation
 

As sint and cost are periodic function with period 2π
So
Sinwt=sin(wt+2π)=sin(t+2π/w). Similarly for cost . 
Hence period of given function if 2π/w
===> D is correct

Q.03 \inline \fn_cm A= (-4x-3y+az)i+(bx+3y+5z)j+(4x+cy+3z)k is irrotational if a,b,c are__________

A.   4,-3,5

B.   4,5,-3

C.   -3,4,5

D.   2,3,5

Check Answer
D
View Explanation
 

Flow is irrational if \fn_cm \nabla \times A=0\\ \\ \begin{bmatrix} i& j & k\\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ (-4x-3y+az)& (bx+3y+5z) & (4x+cy+3z) \end{bmatrix}=0\\ \\(c-5)i-(4-a)j+(b+3)k=0i+0j+0k\\ \\ (a,b,c)=(4,-3,5)

Q.04 \fn_cm V= (-4x-6y+3z)i+(-2x+y-5z)j+(5x+6y+az)k  is solenoidal if a = ________

A.   1

B.   2

C.   3

4.   4

Check Answer
C
View Explanation
 

For solenoidal vector  \fn_cm \nabla.V=0
After applying partial derivative we get
-4+1+a=0
So a=3
C is correct

Q.05 \fn_cm \fn_cm \int_{(0,0)}^{(2,1)} (10x^4-2xy^3)dx-3x^2y^2    along the path \fn_cm x^4-6xy^3=4y^2 is

A.   56

B.   60

C.   62

D.  64 

Check Answer
B
View Explanation
 

We know that 

\fn_cm M=(10x^4-2xy^3),N=-3x^2y^2\\ \\ \\ \frac{\partial M}{\partial y}=-6x^2\\\\\\ \frac{\partial N}{\partial x}=-6x^2

As the solution is exact. So we have

\fn_cm \int_{(0,0)}^{(2,1)} (10x^4-2xy^3)dx-independent \; terms \; of \; x\\\\\\ \int_{(0,0)}^{(2,1)} (10x^4-2xy^3)dx-0 \\\\\\ ={2x^5}-x^2y^3|_{(0,0)}^{(2,1)}\\\\\\ ={64}-{4}=60

Q.06 If S is the closed surface and v is the volume enclosed by S then \fn_cm \int \int_s r.nds=
A. v
B.  2v
C.  3v
D.  4v

Check Answer
C
View Explanation
 

By Gauss Divergence Theorem 

\fn_cm \int \int_s F.\widehat{N}ds=\int \int_r \int dN.Fdv

As position vector \fn_cm \overrightarrow{r}= x i+yj+zk

So  \fn_cm \int \int_s r.nds=\int \int_v \int \nabla.\overrightarrow{r}dv\\\\\\ =\int \int_v \int (\frac{\partial }{\partial x}i+\frac{\partial }{\partial y}j+\frac{\partial }{\partial z}k).(xi+yj+zk)dv\\\\\\ =\int \int_v \int(1+1+1)dv\\\\\\ =3\int \int_v \int dv =3v

Q.07 Centrifugal acceleration is

A.  \fn_cm -\omega \times (\omega \times r)

B.  \fn_cm \omega \times (\omega \times r)
C.  \fn_cm \omega . (\omega \times r)
D.  \fn_cm r \times (\omega \times r)

Check Answer
A
View Explanation
 

As \fn_cm \overrightarrow{r}= cos \omega t + sin\omega t\\\\\\ \frac {d \overrightarrow{r}}{dt}=\overrightarrow{v}=- \omega sin \omega t +\omega cos\omega t\\\\\\ \frac {d \overrightarrow{v}}{dt}=\overrightarrow{a}=- \omega^2 cos \omega t -\omega^2 sin\omega t\\\\\\ \overrightarrow{a}=-\omega^2 \overrightarrow{r}=-\omega \times( \omega \times \overrightarrow{r} )

Q.08  Number of degrees of freedom of two particles connected by a rigid rod moving treely in a plane Is

A. 2
B.  3
C.  4
D.  5

Check Answer
C
View Explanation
 

No. of degree of freedom= No. of particles × 2

                                              = 2  ×  2  =   4

Q.09  The centrold of a uniform semicircular wire of radius a is

A. \fn_cm \frac{2a}{\pi}

B.  \fn_cm \frac{4a}{\pi}

C.  \fn_cm \frac{a}{\pi}

D.  \fn_cm \frac{a}{2\pi}

Check Answer
A
View Explanation
 

In processd

Q.10  Moment of inertia of a rectangular plate with sides a,b about an axis perpendicular to plate and passing through vertix is

A. \fn_cm \frac{1}{3}Ma^2

B.  \fn_cm \frac{1}{3}Mb^2

C. \fn_cm \frac{1}{3}M(a^2-b^2)

D.  \fn_cm \frac{1}{3}M(a^2+b^2)

Check Answer
D
View Explanation
 

In process

PUNJAB PUBLIC SERVICE COMMISSION WRITTEN TEST FOR THE POST OF LECTURER IN MATHEMATICS 2015

This Post Has One Comment

  1. Muhammad Imran ghallo

    MashAllah Very nice work. Sir Please answers jaldi upload kr do.

Leave a Reply